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Interesting Triangle

Gregory Taylor sends this along, and asks a really great question about the work:

“Look past the problem of the original triangle having no 90 degrees… they know enough to run a (problematic) check on height to investigate ambiguity of sine. Why would they even do that if they thought it was a right triangle?”

Any answers?

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I haven’t answered any of these before, because I don’t know how to help a student I haven’t met. But Gregory, I don’t think “they know enough to run a (problematic) check on height to investigate ambiguity of sine.” I think the student saw it on the board enough times that they thought it was ‘the next step’.

I’m pleased that they sketched the triangle at the beginning. The angle looked right-ish, so maybe some of the assumptions in the work aren’t too crazy. If they had the habit of re-drawing the picture at the end, maybe they’d see that there’s no way these numbers could be even close to correct … though probably they would be more suspicious of the later parts of the work than the initial computation of the length of the third side, so it might still be tough for them to discover where their thinking went wrong.

Some context for you then – this was actually a high achieving student, who you can see is even careful enough to use root(164) rather than rounding off to decimals as soon irrationals appear. One of their main issues is they got so intuitive in their approaches, they rarely wrote things down or second guessed what they were doing.

I guess Joshua has a point in being suspicious only of the later work. Part of why I called the check “problematic” is that they SEEMED to use sine, got an answer that didn’t jibe with what they’d already found, and assumed the CHECK was wrong, rather than their original idea of a right triangle. (Thus switched it to Cosine.) I admit to guessing there.

Still, why not back up to the 33 degrees, when first of all it’s simultaneously drawn like 90, and secondly I try to hammer home LONGEST SIDE always opposite LARGEST ANGLE? Any extra thoughts?

Yeah, that last paragraph of yours is the one, Gregory. It’s the question of how do we get kids to check their work seriously, as part of doing the work, rather than as a formulaic thing that’s just there because teachers make them do it?

For me it was easy — learning to check my work constantly (at steps along the way, and not just a final answer) was the way I could keep up with some of my more talented math buddies in school (this was around middle school age). They might make fewer mistakes, but if I could find all of mine as I was going along and fix them promptly, then I’d do OK. And now it’s so much of a habit that I sometimes check my work in different ways more than a dozen times a minute while I’m doing a complicated algebra problem. Stuff like “what’s the last digit, should that be bigger or smaller than the other thing, is that ratio still approximately 2:1” … those kinds of checks are constantly on my mind while I’m going through any long piece of work.

So, how do I convince students that it’s worthwhile to devote that big a fraction of the mental energy of doing the problem to checking whether things make sense?

I wish I knew the answer. I’ve done all kinds of attempts at it. One that’s worked for some kids is getting them to spend time guessing at the beginning, and then explaining after they do their work something about whether the guess was close or way off, and if it was way off, why. That at least gets the “sanity check” part.

Here, I’d be expecting to see a sketch of the two SSA possibilities at the beginning, drawn a lot bigger and a little closer to scale than the one your student provided, so that they could see that there should be two values for A and at least roughly how big it is. My mental estimate is “one option should be just a bit more than 0, and the other one is, well, 8 is almost 10 so it’s almost isosceles and that leaves almost 140 degrees, a little les since the angle opposite the 10 will be more than the angle opposite the 8”. So then my answer should match that “guess” reasonably closely.